Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity

Abstract

We show that the n-fold integrals ${\chi }^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, are actually diagonals of rational functions. As a consequence, the power series expansions of these solutions of linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. Besides, in a more enumerative combinatorics context, we show that generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We give a large set of results illustrating the fact that the unique analytical solution of Calabi-Yau ODEs, and more generally of MUM ODEs, is, almost always, diagonal of rational functions. We revisit Christol's conjecture that globally bounded series of G-operators are necessarily diagonals of rational functions. We provide a large set of examples of globally bounded series, or series with integer coefficients, associated with modular forms, or Hadamard product of modular forms, or associated with Calabi-Yau ODEs, underlying the concept of modularity. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity (in particular integrality of the Taylor coefficients of mirror map), introducing new representations of Yukawa couplings.

arXiv comment100 pages

References in corpus (17)

• 25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple cubic lattice
• Arithmetic Properties of Mirror Map and Quantum Coupling
• On reducing the Heun equation to the hypergeometric equation
• Tables of Calabi--Yau equations
• On Rationally Parametrized Modular Equations
• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• High order Fuchsian equations for the square lattice Ising model: ${\tilde{\chi }}^{(5)}$
• Square lattice Ising model susceptibility: connection matrices and singular behavior of ${\chi }^{(3)}$ and ${\chi }^{(4)}$
• Square lattice Ising model susceptibility: Series expansion method and differential equation for ${\chi }^{(3)}$
• A library of extended high-temperature expansions of basic observables for the spin S Ising models on two- and three-dimensional lattices
• The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
• Transformations of some Gauss hypergeometric functions
• The diagonal Ising susceptibility
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• New Algorithm of the Finite Lattice Method for the High-temperature Expansion of the Ising Model in Three Dimensions
• Explicit formula for the generating series of diagonal 3D rook paths
• The modularity conjecture for rigid Calabi-Yau threefolds over Q

Cited by in corpus (7)

• Creative Telescoping for Holonomic Functions
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
• Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics
• Counting With Irrational Tiles
• Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version)
• Automata and the susceptibility of the square lattice Ising model modulo powers of primes